Nonlinear functional estimation: Functional detectability and full information estimation

We consider the design of functional estimators, i.e., approaches to compute an estimate of a nonlinear function of the state of a general nonlinear dynamical system subject to process noise based on noisy output measurements. To this end, we introduce a novel functional detectability notion in the form of incremental input/output-to-output stability ($\delta$-IOOS). We show that $\delta$-IOOS is a necessary condition for the existence of a functional estimator satisfying an input-to-output type stability property. Additionally, we prove that a system is functional detectable if and only if it admits a corresponding $\delta$-IOOS Lyapunov function. Furthermore, $\delta$-IOOS is shown to be a sufficient condition for the design of a stable functional estimator by introducing the design of a full information estimation (FIE) approach for functional estimation. Together, we present a unified framework to study functional estimation with a detectability condition, which is necessary and sufficient for the existence of a stable functional estimator, and a corresponding functional estimator design. The practical need for and applicability of the proposed functional estimator design is illustrated with a numerical example of a power system.